page_title: Exponential type
hierarchical_section_title: Exponential type
caption_attribution_description: English: In gray is the Gaussian Exp[-\pi *x^2], the red function is an extremal majorant, the blue function is an extremal minorant, each of exponential type 2\pi. The partial sums here are taken symmetrically to 100.
context_page_description: In complex analysis, a branch of mathematics, a holomorphic function is said to be of exponential type C if its growth is bounded by the exponential function eC|ᶻ| for some real-valued constant C as |z| → ∞. When a function is bounded in this way, it is then possible to express it as certain kinds of convergent summations over a series of other complex functions, as well as understanding when it is possible to apply techniques such as Borel summation, or, for example, to apply the Mellin transform, or to perform approximations using the Euler–Maclaurin formula. The general case is handled by Nachbin's theorem, which defines the analogous notion of Ψ-type for a general function Ψ as opposed to eᶻ.
context_section_description: In complex analysis, a branch of mathematics, a holomorphic function is said to be of exponential type C if its growth is bounded by the exponential function eC|ᶻ| for some real-valued constant C as |z| → ∞.  When a function is bounded in this way, it is then possible to express it as certain kinds of convergent summations over a series of other complex functions, as well as understanding when it is possible to apply techniques such as Borel summation, or, for example, to apply the Mellin transform, or to perform approximations using the Euler–Maclaurin formula.  The general case is handled by Nachbin's theorem, which defines the analogous notion of Ψ-type for a general function Ψ(z) as opposed to eᶻ.
caption: The graph of the function in gray is 
    
      
        
          
            e
            
              −
              π
              
                z
                
                  2
                
              
            
          
        
      
      {\displaystyle e^{-\pi z^{2}}}
    
  , the Gaussian restricted to the real axis. The Gaussian does not have exponential type, but the functions in red and blue are one sided approximations that have exponential type 
    
      
        
          2
          π
        
      
      {\displaystyle 2\pi }
    
  .
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